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Breaking a Graph into Connected Components with Small Dominating Sets

Authors Matthias Bentert, Michael R. Fellows , Petr A. Golovach , Frances A. Rosamond , Saket Saurabh

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Author Details

Matthias Bentert
  • University of Bergen, Norway
Michael R. Fellows
  • University of Bergen, Norway
  • Lebanese American University, Beirut, Lebanon
Petr A. Golovach
  • University of Bergen, Norway
Frances A. Rosamond
  • University of Bergen, Norway
  • Lebanese American University, Beirut, Lebanon
Saket Saurabh
  • The Institute of Mathematical Sciences, Chennai, India

Funding

  • Bentert, Matthias: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).
  • Fellows, Michael R.: Supported by the Alexander von Humboldt Foundation under the Humboldt Renewed Research Stay in Germany and the Return to Germany Program for Humboldt Research Prizewinner Fellows.
  • Golovach, Petr A.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Rosamond, Frances A.: Supported by the Western Sydney University and the Alexander von Humboldt Foundation under the Humboldt Renewed Research Stay in Germany for Humboldt Research Prizewinner Fellows.
  • Saurabh, Saket: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416).

Cite As Get BibTex

Matthias Bentert, Michael R. Fellows, Petr A. Golovach, Frances A. Rosamond, and Saket Saurabh. Breaking a Graph into Connected Components with Small Dominating Sets. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.24

Abstract

We study DOMINATED CLUSTER DELETION. Therein, we are given an undirected graph G = (V,E) and integers k and d and the task is to find a set of at most k vertices such that removing these vertices results in a graph in which each connected component has a dominating set of size at most d. We also consider the special case where d is a constant. We show an almost complete tetrachotomy in terms of para-NP-hardness, containment in XP, containment in FPT, and admitting a polynomial kernel with respect to parameterizations that are a combination of k,d,c, and Δ, where c and Δ are the degeneracy and the maximum degree of the input graph, respectively. As a main contribution, we show that the problem can be solved in f(k,d) ⋅ n^O(d) time, that is, the problem is FPT when parameterized by k when d is a constant. This answers an open problem asked in a recent Dagstuhl seminar (23331). For the special case d = 1, we provide an algorithm with running time 2^𝒪(klog k) nm. Furthermore, we show that even for d = 1, the problem does not admit a polynomial kernel with respect to k + c.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Parameterized Algorithms
  • Recursive Understanding
  • Polynomial Kernels
  • Degeneracy

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